Horváth, E.; Puczyłowski, E. R. Some remarks on simple rings. (English) Zbl 0725.16007 Boll. Unione Mat. Ital., VII. Ser., A 5, No. 1, 83-87 (1991). Let a subring S of a ring R be accessible by one-sided ideals, \(S^*\) the ideal of R generated by S, and \(\beta\) the prime radical. It is proved that S/\(\beta\) (S) is simple if and only if \(S^*/\beta (S^*)\) is simple. Moreover, \(\beta (S)=\{s\in S:\) \(S^ msS^ n\subseteq \beta (R)\}\) where m and n are integers with \(S^ mRS^ n\subseteq S\) (such m and n exist whenever S is one-sided accessible). Applying the results the authors get easily results of P. N. Stewart and J. F. Watters [Math. Chron. 16, 85-87 (1987; Zbl 0646.16031)] on simple rings and Morita contexts. For any \(0\neq a\in R\), let R(a) be the ring with addition as in R and multiplication \(r_ 1*r_ 2=r_ 1ar_ 2\) for all \(r_ 1,r_ 2\in R\). If \(Ra=(Ra)^ 2\), then R(a)/\(\beta\) (R(a)) is simple if and only if RaR/\(\beta\) (RaR) is simple. This generalizes a theorem of U. Hirzebruch [Arch. Math. 20, 578-579 (1969; Zbl 0215.090)]. Reviewer: R.Wiegandt (Budapest) MSC: 16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras 16N60 Prime and semiprime associative rings Keywords:subring; prime radical; one-sided accessible; simple rings; Morita contexts Citations:Zbl 0646.16031; Zbl 0215.090 PDFBibTeX XMLCite \textit{E. Horváth} and \textit{E. R. Puczyłowski}, Boll. Unione Mat. Ital., VII. Ser., A 5, No. 1, 83--87 (1991; Zbl 0725.16007)